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Improving the prediction of zonal modeling for forced convection airows inrooms

M.O. Abadiea,*, M.M. de Camargob, K.C. Mendonab, P. Blondeaua

a LEPTIAB, University of La Rochelle, Avenue M. Crpeau, 17042 La Rochelle Cdex 1, Franceb Mechanical Engineering Department, Pontifcia Universidade Catlica do Paran e PUCPR, Rua Imaculada Conceio, 1155, Prado Velho, 80215-901 Curitiba, PR, Brazil

a r t i c l e i n f o

Article history:

Received 16 March 2011

Received in revised form

5 September 2011

Accepted 7 September 2011

Keywords:

Zonal

Modeling

Forced convection

Ventilation

Airow

Room

a b s t r a c t

Evaluating the efciency of a ventilation system in providing a healthy indoor environment to buildingoccupants requires the knowledge of the airow inside the room prior to the calculation of the pollutant

dispersal from their sources. Computational Fluid Dynamics (CFD) modeling is probably the most suit-able method to achieve this as it provides detailed information about the airow pattern but is still time-

consuming although processor speed has tremendously increased during the past years. Consequently,parametric studies aimed at characterizing the inuence of parameters such as the location of the air

inlets and outlets or furniture on the air velocity, temperature and concentration distributions within thebuilding are seldom achieved. Zonal, Coarse-grid CFD and Fast Fluid Dynamics (FFD) are intermediate

models between CFD and single air node models; they can predict the air ow pattern in a room or groupof rooms with less computational efforts but with a lower accuracy than CFD models. The present study

aims at improving the zonal prediction for the case of forced convection in indoor spaces. Results showthat the proposed alterations of zonal modeling greatly improve the prediction and that this model

requires less computational efforts and returns more accurate results than Coarse-grid CFD and FFD. 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Nodal models such as ENERGYPLUS [1], TRNSYS [2] and ESP-r [3]for energy calculations or CONTAM[4], IAQX[5]and IA-QUEST[6]for indoor air quality analyses, to cite a few, are widely used in the

building simulation community. Those codes allow a fast evalua-tion of energy consumption, thermal comfort, pollutant transportand exposure in indoor environments; valuable information in the

search for low energy consumption and healthier buildings.However, they are based on the strong assumption that statevariables (temperature, relative humidity, pollutant concen-

tration.

) are homogeneous within the simulated zone; well-mixed assumption that proves too simplistic for modeling largevolumes where thermal stratication takes place, for estimatingthermal comfort where temperature and air velocity at the occu-pant location are needed and for the initial period of the mixing of

a pollutant in the room air, particularly for point source releases.Coupling those programs with Computational Fluid Dynamics(CFD) that solves the equations expressing the conservation of

mass, momentum and energy has been seen as a logical solution to

improve the nodal program prediction by acquiring additionalinformation about the variable heterogeneity that really occursindoors [7,8]. If CFD methods can in principle provide thesenecessary details, they cannot yet, practically, be applied to long-

time period (e.g. seasonal or annual) analyses nor to largecomplex buildings because they still remain time consuming bothin preparation and simulation running.

Intermediate modeling methods between nodal and CFD doexist. Especially, the airow, temperature and possibly contaminantdistributions within rooms can be predicted using the zonal

approach, a method that combines the advantages and weaknessesof the nodal and CFD approaches. Zonal modeling consists rst individing the roomvolume into a small numberof zones or cells, andthen to assume perfect mixing of the air within each zone:temperature, humidity and possibly contaminant concentrations

are considered homogeneous, but pressure varies hydrostatically.The airow and temperature distributions are determined simul-taneously by solving the mass and energy balance equations ineach zone. A simplied expression of the momentum equation is

used instead of the NaviereStokes equation, which has theadvantage of simplicity, but faces the problem that the effects ofdriving ows such as boundary layer ows, thermal plumes and* Corresponding author. Tel.: 33 5 46458624; fax: 33 5 46458241.

E-mail address: mabadie@univ-lr.fr(M.O. Abadie).

Contents lists available atSciVerse ScienceDirect

Building and Environment

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c om / l o c a t e / b u i l d e n v

0360-1323/$ e see front matter 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.buildenv.2011.09.006

Building and Environment 48 (2012) 173e182

mailto:mabadie@univ-lr.frhttp://www.sciencedirect.com/science/journal/03601323http://www.elsevier.com/locate/buildenvhttp://dx.doi.org/10.1016/j.buildenv.2011.09.006http://dx.doi.org/10.1016/j.buildenv.2011.09.006http://dx.doi.org/10.1016/j.buildenv.2011.09.006http://dx.doi.org/10.1016/j.buildenv.2011.09.006http://dx.doi.org/10.1016/j.buildenv.2011.09.006http://dx.doi.org/10.1016/j.buildenv.2011.09.006http://www.elsevier.com/locate/buildenvhttp://www.sciencedirect.com/science/journal/03601323mailto:mabadie@univ-lr.fr

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jets cannot be handled properly. Consequently empirical and semi-empirical equations representing those driving ows must be

added to the problem. Although the zonal method has been used ina large number of applications during the last three decades [9],a fully validated model considering the most common driving ows

in buildings is still lacking. Most studies have addressed naturalconvection problems and concluded that the zonal approach isrelevant for such problems[10e16]. On the other hand, few studieshave dealt with the validation of zonal models in the context of

mechanically-ventilated rooms or enclosures. Zonal modelsconsidering a 2D jet model[13,17,18]or a 3D jet model [18]weretested against experimental data measured in a square [19] or

a rectangular cell[20]which was ventilated with a horizontal walljet. All those studies concluded that the jet region was much betterpredicted than the recirculation zone of the room. However, it mustbe noted that only a few number of experimental data were

available, and that some comparisons were performed based onmeasured velocity proles while zonal models simply return themean velocity of the air passing through the interface between twoadjacent zones.

Another intermediate modeling method for airow predictionin indoor environments is Coarse-grid CFD. Compared to standard

CFD, this method can signicantly reduce computational require-ments by solving the RANS equations with a signicantly lowerspatial and temporal resolution[21]. The aim of such a Coarse-gridsimulation is to retain grid-independence in terms of the qualita-tive features of theow whilst accepting a higherlevel of numerical

error. When reviewing the uid mechanics of natural ventilation,Linden [22] suggested that Coarse-grid CFD could be a possiblesubstitute for zonal modeling. Mora et al. [17] concluded thatcoarse-grid CFD returns more accurate results than zonal modeling

for the case of forced convection in rooms. On the other hand, ina more recent study, Gobeau and Deevy [23] gotsimilar results withCoarse-grid CFD and zonal modeling when simulating the decay ofa pesticide concentration in a naturally ventilated room.

Fast Fluid Dynamics (FFD) simulation is the third intermediate

approach available to compute air ows and resulting distributionsof temperature and pollutant concentration in indoor environ-ments. The FFD solves the NaviereStokes equation as the CFD does

but, as the computing speed is the most important issue in FFDsimulation, the FFD uses simple and low-order schemes to reducethe computing cost[24]. For instance, it uses the linear interpola-tion in the semi-Lagrangian approach instead of a high-order

nonlinear interpolation. The pressure-correction projectionmethod used by the FFD is also the simplest among the differentprojection schemes. Finally, another characteristic of the FFD is thata low-order discretization scheme (rst order for time and second

order for space) is applied, but with the associated potential risk togenerate too much numerical dissipation[25]. As a consequence,the FFD has a lower computing cost, but a lower accuracy than the

CFD. FFD was compared to experimental data for cases of fullydeveloped turbulent ow in a plane channel, forced convectionow in a room, natural convection ow in a tall cavity and mixed

convection ow in a room[24,26].Results show that FFD performsreasonably well for all cases except for the forced convection owin a room where large discrepancies are observed.

The present study deals with the improvement of zonal modelsfor the case of forced convection in mechanically ventilated rooms.Section 2 of the paper presents the equations dening the referencezonal model for this case, as well as the experimental setup and

data to be used as a benchmark. Based on the comparison betweennumerical and experimental results, alterations have been made tothe model and simulations were repeated as a way to assess theimprovements in the airow prediction (Section 3). Finally, the

improved zonal model has been evaluated against other existing

zonal models as well as Coarse-grid CFD and FFD. The results fromthis comparative analysis are presented in Section4.

2. Reference zonal model

Based on the principles of zonal modeling presented above, thereference zonal model for the case of forced convection will bemade of two kinds of elemental cell models: the standard cellmodel that represents the ow regions with weak momentum and

the jet cell model that describes the air movement in the jet region.Here, the room air volume was divided into rectangular parallele-pipeds as a way to generate a simple structured mesh grid.

2.1. Standard cell model

Under the assumption that the air is perfectly mixed within

cells, the heat and dry air mass conservation equations can beexpressed as follows:

ricpVidTidt

_Qsource;i X6

j 1

_Qij X6

j 1

fij (1)

X6j 1

_mij 0 (2)

wherer i is the air density (kg/m3),cpthe specic heat (J/kg.K), Vi

the cell volume (m3),Tithe absolute temperature (K), _Qsource;i theheat source or sink (W). _Qijand fij(W) represent the diffusive andadvective heat ux across the interface separating cells i and j,

respectively; diffusion actually proves to be negligible compared toadvection in many situations. _m, the airow rate (kg/s) across theinterface separating cell i from one of its six adjacent cells j, is

considered to be a function of the pressure difference between thecells. Assuming steady-state, uniformity, low velocities (less than0.25 m/s), and further considering that pressure is the unique

surface force and gravity is the unique volume force, the Bernouilliequation leads to the following expressions of _m for vertical andhorizontal interfaces, respectively:

_mij CdAijri

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pi pjrmoy

s ifpi pj and

_mij CdAijrj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pj pirmoy

s ifpi

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pi riRaTi (5)

withRabeing the air constant (J/kg.K).

2.2. Jet cell model

All jet cells are divided into two sub-cells: one is described by

equations that are specic to the type of jet, while the other isdescribed by the standard cell model presented above. Together,the two sub-cells form a rectangular cuboid cell which is used for

the room volume meshing as illustrated inFig. 1.In the context of the present study, equations for isothermal

two-dimensional wall plane jets were considered as the jet sub-cellmodel [27]. The jet characteristics thickness, throw and total

airow rate, are then expressed by Eqs. (6), (7) and (8), respectively.

dx CUx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln 0:01

0:937

r 0:14

! (6)

xjet h0KvU00:25

2

(7)

_mx _m0

1:054CUKv

ffiffiffiffiffiffix

h0

r (8)

Whered(x) is the jet thickness (m),xjetis the jet throw (m), _mx is

the airow rate (kg/s), h0 is the inlet height (m), U0 is the airvelocityat the inlet (m/s),CU 0.068 andKv 3.5. Note that the origin forxis ctitious and can be calculated evaluating x0 fromdx x0 h0 for the thickness and _mx x0 _m0 for the

airow rate.Physically, the jet and standard sub-cells are linked together by

imposing the entrainment airow rate between the cells. Theentrainment airow in the jet cell is evaluated as the difference

between the total airow rates at the upstream and downstream

interfaces using Eq.(8). The airow direction is also imposed to betoward the jet core at the external interfaces of the standard sub-

cells as a way to prevent an incorrect velocity direction betweenthe jet and entrainment zones, as it was observed in[28]. Practi-cally, this is achieved by disabling the connections between thestandard sub-cells along the jet expansion.

2.3. Simulation environment

The present zonal model has been integrated in TRNSYS which isa simulation environment for transient simulation of systems,including multi-zone buildings[2]. The source code of the kernel aswell as the component models is delivered to users. A TRNSYS

project is typically setup by connecting components called Types

graphically in the Simulation Studio. Each Type is described bya mathematical model in the TRNSYS simulation engine and hasa setof matching Proformas in the Simulation Studio. The Proforma

has a black-box description of a component: inputs, outputs,parameters, etc. In this way, two new TRNSYS Types representingthe standard and jet cells along with their documented Proformas

have been developed in FORTRAN. Connections between standardcells are made sharing the pressure, temperature and cell height(that is needed to calculate the hydrostatic term of Eq. (4)). Linksbetween jet cells and others are made by the airow rate value.

According to the values of the neighboring cells, the pressure andthe temperature of a considered cell are calculated iteratively usingthe NewtoneRaphsonmethod as described in COMIS fundamentals[29]. The convergence of the whole problem is performed by

TRNSYS using the successive substitution algorithm (root-mean-

square residuals lower than 106). With this iterative process, theoutputs of a given Type are substituted for the inputs of the nextType in the system. This substitution continues at a given time stepuntil all connected outputs have converged.

2.4. Test case: Nielsens experiment

The ventilated cavity test case described in [20]was chosen toevaluate the reference zonal model. This test case considersisothermal conditions (20 C) in a mechanically ventilated room of

9 m long (L), 3 m wide (W) and 3 m high (H) ( Fig. 2). The air issupplied horizontally on the upper left corner of the room througha 0.168 m high (h0) air inlet having a variable width v, and isexhausted through an opening located on the lower right corner, on

the opposite side to the air inlet. This air outlet is 0.48 m high (t)and 3 m wide. The inlet conditions for the velocity were taken as

U0 0.455 m/s, which gives an inlet height-based Reynoldsnumber, Reh0, of 5000. Moreover, the conguration where the air

inlet had the same width as the room (v 3 m) was considered for

the purpose of this study. For these conditions, the side wallscan beassumed to have a negligible inuence on the airow conditions in

the central part of the room. Hence the problem can be treated astwo-dimensional.

Nielsens test case was rst studied experimentally: horizontalvelocity measurements have been made along two vertical line in

x H andx 2H in the central plane (z 0.5 W). It has also beenextensively taken as a benchmark test for airow models since it isrepresentative of usual ventilated room. As shown by Susin et al.[30], the jet expands in the upper part of the room, creating

a recirculation in the middle of the room volume (Fig. 3). Velocitiesare higher near the wall opposite to the inlet wall and close to the

oor. Based on the equations for an isothermal wall plane jet (Eqs.(6) and (7)), the jet throw and thickness are 6.82 m and 1.26 m,

respectively. Besides, it is noticeable that the jet throw obtained forReh0 5000 is the maximum length for a jet in this room. Indeed,

Fig. 1. Description of a jet cell.

Fig. 2. Description of Nielsen

s experiment setup.

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Awbi and Setrak [31] derived a theoretical expression of the

distance from the supply of a plane wall jet for which the effect ofthe opposite wall is not detectable, xf, as a function of the supplyopening height,h0, and the distance of the wall from the supply, L:

xfh0

0:52

L

h0

1:09(9)

For the present conguration, Eq.(9)returnsxf 6.70 m, whichmeans that forhigher Reh0, thejet throw would be roughly thesame,and thus the structure of the airow in the room would remain

unchanged. This point had been demonstrated experimentally byNielsen et al.[32]who reported that the dimensionless proles areindependent of Reynolds numberwhen the latterranges from 5000to 10000. As a denitive validation, it can nally be noted that thereference zonal model has been tested for Reh0 10000; the results

showed unaltered dimensionless velocity proles.

3. Evaluation and improvement of the zonal model

xjet 6.82 m and djet 1.26 m (Fig. 3) determine the region of

the room that has to be modeled as a jet, whereas the remainingvolume can be treated using standard cell models. Here, a 6 3computational grid including 5 jet cells of 1.5 m long and 1.5 m high(noted JET inFig. 4a) and 13 standard cells was designed. The room

volume has been divided into 6 different cells in the direction of thejet, although 2 would have been enough. The reason is thatexperimental and CFD data are available at abscissa x 3 m and

x 6 m. Therefore, that way, the vertical interfaces between cells

coincide with the benchmark, which makes the model evaluationeasier. On the other hand, the zonal jet region (7.5 m by 1.5 m) isslightly greater than the theoretical jet dimension using this mesh,

which may result in some errors. However, these errors can beexpected to be insignicant compared to the uncertainties coming

from the zonal method itself. Fig. 4a and b present the dimen-sionless velocities at the cell interfaces that were computed by 2DCFD and the zonal model as described above, respectively. Acommercial CFD program [33] has been used for numerical

prediction. The governing equations are solved with a segregatedscheme and a second order high resolution advection scheme hasbeen adopted. The continuity equation is a second order centraldifference approximation to the rst order derivative in velocity,

modied by a fourth derivative in pressure which acts to redis-tribute the inuence of the pressure. The standard k-eps turbulentmodel of Launder and Spalding[34]with a structured mesh gridmade up of 44 36 cells with grid renement in the jet region and

at the walls has been used to assure the correct calculation of thewall boundary layer by the wall function (20 < y

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rate as the quadratic sum of the two ow rates. This method, thatwill be referred to as zonal/jet in the next sections, is a commonlyused procedure to handle the cumulated effects of wind andthermal forces when dealing with natural ventilation and inltra-tion problems in buildings [35]. Numerically, it has been imple-

mented by modifying the mass ow rate equation introducinga constant,K, and solving the following equation:

_ment:

x ljet

2

_mrec:x ljet

2

_mx Kljet

2(10)

Considering the maximum values of entrainment and recircu-lation mass ow rates:

max

_ment:

x ljet max

_mrec:

x ljet _m

x ljet

(11)

From Eq.(8),Kis evaluated as:

K 2 h0

ljet1:054CUKv2

(12)

The mass ow rate for isothermal wall plane jet in connedspace is then given by:

_mx _m0

1:054CUkv

ffiffiffiffiffiffiKx

h0

s ! (13)

Fig. 6 Isothermal wall plane jet e Quadratic sumcurve shows

that the new jet model much better predicts the increase in the

mass ow rate with distance to the inlet. Similar trend can beobserved onFig. 4c, which presents the dimensionless velocities atthe cell interfaces. However, Fig. 4c also shows that the imple-

mentation of the new jet model does not signicantly change thepredicted airow out of the jet region. Especially, air velocities nearthe oor are still too low compared to those predicted by CFD.

This problem originates from the absence of momentum equa-tion in the zonal method. Consequently, the mass ow rates outsideof the jet region tend to be uniformly distributed over the roomvolume whereas dominant ow paths exist. As expressed by Eqs.

(4) and (5), the mass ow rate between two adjacent standard cellsis determined by the pressure difference between the two cells andthe ow coefcientCd. Usually, the same ow coefcient is appliedto all cell interfaces, and the problem is to set the appropriate value

for this coefcient. Here however, the boundary conditions of thestandard cells zone (i.e. the jet interfaces) are dened in terms of

xed ow rates. Therefore, the mass ow rates between the stan-dard cells are independent of the ow rate coefcient. One possiblesolution to adjust theow rates between standard cells is to change

the ow coefcients depending on the location of the cells.Previous studies have addressed this point. Axley[36]varied boththe ow coefcient and power-law viscous loss of Eqs. (4) and (5).In particular, the Power-Law with Mixing Length approximation

model (PLML) that he proposed considers different ow coef-cients depending on the distance from the walls. Axley [36]alsocalled to Nielsens experiment as a reference test case to evaluatethe resulting zonal model and concluded that it can improve the

airow prediction, especially near the wall opposite to the inlet.

Nevertheless, by comparing zonal and coarse-grid CFD models,Mora et al.[37]came to the conclusion that Axley s model does not

provide a better representation of the air recirculation within theroom than the reference zonal model (i.e. considering same owcoefcients for all standard cells) does. Another approach has been

rst investigated by Mora [38] and subsequently developed by

Teshome and Haghighat[39]. It consists in inversing the problemby evaluating the ow coefcients based on the pressure distri-bution determined by CFD simulation. Mora[39]computed those

coefcients for Nielsens reference test case (Reh 5000) and thenused zonal modeling to predict the airow for different Reynolds

numbers. He showed only small discrepancies between thepredictions and the experimental results. Teshome and Haghighat[40] used the same procedure but they splitthe ow coefcient into

a vertical and a horizontal component. They showed that theresulting variable-ow-coefcient zonal model contributes toa signicantly better prediction of both the ow pattern andmagnitude of air velocities compared to the reference zonal model.

One shortcoming of this methodology is nevertheless that the owcoefcients strongly depend on the geometry of the problemaddressed; for instance, the ow coefcients may have verydifferent values if considering other locations of the air inlet. In

other words, the model is not valid for congurations other thanthe one considered for the numerical developments, which isobviously a strong limitation and fundamentally not what can beexpected from a model.

The method proposed in this work to handle the problem isbasically different: instead of considering changes at the micro-scopic level, as previous studies did, a macroscopic approach that

Fig. 6. Evolution of the jet mass

ow rate as a function of the distance to the inlet (x).

Fig. 5. Entrainment for a jet in open space (a) and for a jet in conned space (b).

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probably best ts the philosophy of zonal modeling has beendeveloped. Fundamentally, the principle of the method is to force

the airow to be parallel to the walls by decreasing by a sameamount the ow coefcients of the cell interfaces that are parallelto these walls (Cells 1e14 inFig. 4d). Only the cells neighboring the

walls are of concern, which means that for the division of the roomvolume considered, the ow coefcients related to the interfaces ofall cells located in the central part of the room should be affected(Cells 15e18 inFig. 4d). Those ow coefcients have been multi-

plied by a correction factor CD of 0.45, a value that was determinedfrom CFD modeling.Fig. 4d presents the dimensionless velocities atthe cell interfaces obtained that way. It can be noted that the

computed velocities correctly t the CFD results.The resultspresented in Fig.4d include the contributions of both

the modied jet equation and variation of the airow coefcients.Fig. 7 presents a deeper analysis of the model improvements

resulting from each step of the development, together witha sensitivity analysis of the correction factor CD. Air velocitiesreturned by CFD have rst been integrated along the zonal meshinterfaces as a way to derive the mean air velocities across each of

the 27 cell interfaces dened by the zonal division of the roomvolume. Then, the mean velocities have been compared to those

returned by the zonal model considering the new facilities imple-mented, and different correction factors. As previously explained,the strong decrease in the maximal discrepancy between thereference zonal model and zonal/jet model originates from a moreaccurate prediction of the mass ow rate in the jet zone. On the

other hand, the median discrepancy is slightly increased, reectinghigher discrepancies between the two models in the standard cellregion of the room. On the whole, the modeling of enhanced airentrainment into the jet due to recirculation actually adds to the

accuracy of the zonal model. Finally, as discussed previously,CD 0.45 had been found to be the optimal value of the correctionfactor from simulations of Nielsens test case: using this value asinput of the model, the median discrepancy is divided by a factor

about two. However,Fig. 7also shows that the air velocity distri-

bution is few affected by CD in the range from 0.4 to 0.6. This is aninteresting point in the sense that it indicates that CD values about0.5 can be condently input in zonal models as a way to improve

the simulation of congurations other than the one investigated. Itis nevertheless important to note that the optimal CD will actuallydepend upon the size of the mesh grid. Simulations of Nielsen sexperiment were repeated considering ner mesh grids and

smallerthicknesses of the near-wall cells than the ones presentedonFig. 4. The optimal correction factor to input in the zonal model

was found to be linearly correlated to the average thickness of the

cells neighboring the walls (Fig. 8).Fig. 9 presents the air velocities along 2 vertical axes (x 2H and

x 2.5H) and 2 horizontal axes (y 0.17H and y 0.5H). Resultsreturned by CFD simulation, reference zonal model, zonal/jet

model, zonal/jet model with CD 0.45, but also measured veloci-ties [20] are gured out. The compared air velocity proles atabscissa x 2H demonstrate the relevance of CFD simulation:

predicted air velocities are very close to the measured ones (EXP).On the other hand, the reference zonal model underestimates thevelocity in the jet region (y > 0.5H) and wrongly predict a piston-type ow below the jet zone (y < 0.5H). The modied jet equa-

tion (zonal/jet model) improves the prediction of the airow in thejet region but the piston-type ow below the jet zone is stillinaccurately predicted. This problem is solved when furtherimplementing the ow coefcient correction in the zonal model.

Same trends can be observed at lower abscissasx 0.5H,x H and

x 1.5H. The results returned by the zonal models at x 2.5H canonly be compared to those returned by CFD since no experimentaldata exist for this abscissa. Such comparison is nevertheless inter-

esting as it highlights one of the main drawbacks of traditionalzonal methods. The problem is the modeling of the reverse airow

below the jet region: air velocities are overestimated in the centralhorizontal region of the room (0.17H < y < 0.5H), while they areunderestimated in the oor neighborhood (y < 0.17H). Once again,implementing the airow coefcient correction (CD 0.45) in themodel improves the results. The air velocity proles at ordinates

y 0.17H andy 0.5H lead to the same conclusion, although theproposed zonal model predicts a longer jet than CFD does (negative

velocities forx > 2H, see last graph).

4. Intermediate models benchmarking

The zonal model proposed here has been evaluated againsta variety of other zonal methods, as well as Coarse-grid CFD and

FFD (Table 1). The results were directly taken from the literature

when the authors used Nielsens test case as a benchmark to assesstheir models. Otherwise, simulations have been carried out for thepurpose of the present study. The models/methods are evaluated

based on the velocity proles at abscissax 2H, as well as statistics

Fig. 8. Optimal ow coefcient factor as a function of the average thickness of near-

wall cells.Fig. 7. Relative errors of zonal modeling methods compared to 2D CFD results.

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about the errors on predicted air velocities at the cell interfaces;measured data were taken as the reference here.

Fig. 10presents the dimensionless velocity proles at the cellsinterfaces for the zonal modeling approaches. First, the left graph

illustrates the need of using a jet model and correctly dening thejet region. In fact, the PL w/o JET is unable to predict a jet region and

a recirculation in the room as all velocity are positive i.e. the pre-dicted airow in this region is oriented from left to right. However,

Table 1

Intermediate models for benchmarking.

Intermediate model Name Description Mesh grid

(cells number inx y) (Fig. 1)

Reference

Zonal PL w/ o JET Power -Law model without any jet model implement ed (only standar d

cells are dened to divide the room volume)

6 6 [36]

PL Power-Law model ereference zonal model including Eq.(8)as

the jet model

6 6 [36]

PLML Power-Law including the mixing Length approximation model 6 6 [35]

CBPL Cell Boundary Power Law model 6 6 [35]

QSNA Quarter-sine prole with Newtons Approximation model 6 6 [35]PL mod. 1 Modied Power-Law model calling for the variable ow coefcient

method (see previous section)9 6 [38]

PL mod. 2 Identical to PL mod. 1 9 9 [38]

ZONAL/JET/CD 0.45 Final form of the zonal model considering the modied jetequation (Eq.(13)) and decreased ow coefcients (CD 0.45)

6 3 Present study

FFD FFD/Laminar Fast Fluid Dynamics model with laminar treatment 36 36 [24]

FFD/100nu Fast Fluid Dynamics model with a constant turbulent viscosity

set to 100 times the molecular viscosity

36 36 [24]

FFD/0-Equ Fast Fluid Dynamics model with a 0-equation turbulence model of

Chen and Xu[40]

36 36 [24]

FFD4 Fast Fluid Dynamics model with modied solving method and

forced mass conservation

60 20 [26]

COARSE-GRID CFD CFD/CG 1 Coarse-Grid CFD 6 6 [36]

CFD/CG 2 Identical to CFD/CG 1 10 10 [36]CFD 2D CFD 2D CFD using the standard k-eps turbulent model of Launder

and Spalding[34]

44 36 Present study

Fig. 9. Dimensionless velocity proles at the cellsinterfaces.

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the four other models presented in this graph tend to highlyunderestimate the jet mass ow rate because the jet cells were only

0.5 m high instead of the necessary 1.26 m to capture the entire jetow rate. The right graph presents the results obtained by the

Modied Power-Law model with variable ow coefcients evalu-ating from CFD simulation (PL mod. 1 and 2). This model givesreasonable results but greater errors than the model from thepresent study.

Fig. 11 presents the same proles for the other intermediatemodels. If comparingrst the only FFD results, the left graph showsthat the FFD/100nu and FFD/0-equ perform poorly compared toFFD/Laminar for the case investigated. As an explanation, Zuo and

Chen[24]suggested that solving the turbulence by CFD methods isnot suited for FFD, and subsequently developed dedicated solvingstrategies [26]. FFD4 is the FFD method that returns the moreaccurate results on the whole. However, this model overestimates

the mass ow rate in the jet. As a result, the airow rate near theoor is also overestimated. The air velocity proles returned bycoarse-Grid CFD are presented in the right graph. Better results areobtained with the thinner mesh grid, although the model slightly

underestimates the mass ow rate in the jet.Fig. 12presents the relative errors of the intermediate models

compared to the experimental measurements. It clearly illustratesthe improvement of the zonal model predictions with the proposed

alterations over the other zonal models. The improved zonal modelreturns also more accurate results than Coarse-grid CFD and FFD.

Apart from accuracy, computing time to reach a convergedsolution is another relevant parameter to assess airow models.

This criterion was assessed by comparing the computational timeratio of each intermediate model/method, an index that was

dened by Zuo and Chen[24]as:

N tphysicaltelapsed

; (14)

Where tphysical is the physical time of ow motion and telapsed iselapsed computing time of the simulation. N 1 means thesimulation is real time while a value ofNgreater than unity means

that the simulation is faster than reality. The computational timeratios for CFD simulations and FFD rst generation models (FFD/Laminar, FFD/100nu and FFD/0-Equ) were found in Zuo and Chen[24]. The computational time for the second generation of FFD

models (FFD4) were passed on by the authors. Coarse-Grid CFDand

Zonal simulations have been performed for the purpose of thepresent study. Note that all zonal models presented before haveabout the same computing time. The related computational time

ratios were corrected as a wayto account for the different processorspeed of the computers. Fig. 13shows that for the isothermal 2Dsteady-state problem investigated, the CFD simulation time is the

same as the physical time. FFD rst generation models are from 30to 50 times faster than real time, whereas FFD second generationmodels and Coarse-Grid CFD are about 150 times faster than real

Fig. 11. Dimensionless velocity pro

les at the cells

interfaces (x

2H) for the other intermediate modeling approaches.

Fig. 10. Dimensionless velocity proles at the cellsinterfaces (x 2H) for various zonal modeling approaches.

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time. Finally, the zonal simulation is clearly the fastest: thecomputational time ratio is about 1500, that is to say 10 times faster

than the other intermediate modeling approaches.

5. Conclusion

The present paper aimed at improving airow modeling byzonal methods in the specic case of forced convection ows in

rooms. Two alterations were implemented in a standard zonalmodel as a way to get more accurate predictions of the jet mass

ow rate and air velocities in the near-wall region. Results showthat the new model returns better but also faster results than

Coarse-Grid CFD and FFD methods. Therefore the proposed zonalmodel is probably well suited for problems where the wholebuilding is to be simulated over long times, with some details aboutthe distribution of the predicted state variables. These include

sharp energy analyses of buildings, natural ventilation design,

thermal comfort studies, and assessment of the occupants expo-sure. As a way to achieve this, next developments will be the

coupling of the airow model with heat transfer and contaminanttransport models.

Limitations and uncertainties nevertheless still remain. First ofthem is that the model has been validated for only one room

conguration. The applicability of the correction that was made tothe ow coefcients must be demonstrated for other room geom-

etries, but the literature lacks fully documented reference test casesuch as Nielsens experiment. Then, one reason why the model

returns good results is that Nielsens experiment put into play a 2Dair jet for which dedicated empirical models have been alreadydeveloped. On the other hand, 3D jets should be considered inmany applications, but there is no simple way to determine the jet

expansion in the third direction. 2D air jet models may prove to beinadequate in some situations. Finally, indoor air problems are mostoften anisothermal. Although natural convection problems havebeen successfully solved by zonal models, considering mixed

convection adds one level of complexity. The denition of empiricalmodels for that case, and the development of methods to reachconvergence are denitely the next steps to overcome for a widerdissemination of zonal models.

Acknowledgments

The authors would like to acknowledge the French Agency forInnovation (OSEO) and the French Agency of the Environment and

the Energy Management (ADEME) for funding in the framework ofthe Vaicteur AIR2 project. The authors are also grateful to W. Guofor providing additional data to their publications.

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